Beijing Institute of Mathematical Sciences and Applications

The study of Lie algebras arose out of the study of symmetries. The theory of finite-dimensional Lie algebras, especially the semisimple ones, is by now a classic branch of mathematics, with many classification results in the late 19th and first half of the 20th century. But the restriction to finite-dimensional symmetry algebras is a matter of convenience, not a fact of nature. Throughout the 20th century, different types of infinite-dimensional Lie algebras have been studied and partially classified. In the second half of the 20th century, Kac and Moody independently arrived at a class of well-behaved Lie algebras, of which the finite-dimensional are precisely the well-known semisimple ones, but containing many new infinite-dimensional ones.

In this course we will focus on the ones of affine type, which can be motivated independently from the Kac-Moody setting, and have a very rich representation theory. In addition, since the 1980s certain deformations of these Kac-Moody Lie algebras have been studied, called quantum groups. Again, especially the affine case is interesting, where the theory was driven by questions in quantum integrable systems: in particular what is an algebraic framework for solutions of the Yang-Baxter equation? In the second half of the course we will explain this connection and discuss some precise results. Throughout, we will focus on the sl(2) case to have straightforward examples.